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\title {Mise en place d'un protocole de paiement pour une basicCard}
\institute{Université de Versailles Saint-Quentin-En-Yvelines}
\author{Mathieu Alinc, Jean-Ronan BOURDONNAY, Sylvain Lamisse et Sophie MARIN}
\date{}


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\begin{frame}
\titlepage
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\begin{center}
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\section{introduction}

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%				Slide 1
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\begin{frame}                           
\frametitle{introduction}  
\begin{itemize}
\item  L'importance des cartes bancaires
\item  Problématique : comment assurer la confiance?
\item  L'intérêt des cartes à puces : programmable $\Rightarrow$ cryptographie
\end{itemize}
\end{frame}                             

\section{Signature et certificat}
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%				Slide 2
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\subsubsection{Signature numérique}
\begin{frame}                           
\frametitle{Notion de signature numérique}  
\includegraphics[scale=0.5]{./Images/sign1.eps}\\
Idée maîtresse = lien entre le message et l'auteur
\begin{itemize}
\item Une signature numérique c'est: \\
Un chaîne de caractère qui dépend du contenu du message soumis à la signature.
\item Une signature doit garantir:
	\begin{itemize}
	\item authentification de l'auteur
	\item intégrité du message
	\item non répudiation
	\end{itemize}
\item Système ouvert $\Rightarrow$ cryptographie asymétrique.
\end{itemize}
\end{frame}
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\subsubsection{Protocole DSA}
\begin{frame}                           
\frametitle{Protocole DSA}
$ \approx$ schéma de signature de ElGamal.\\
\includegraphics[scale=0.25]{./Images/Digital_Signature_diagram.eps}\\
\begin{enumerate}
\item Génération des clés 
\item Signature
\item Vérification
\end{enumerate}
\end{frame}
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\begin{frame}                           
\frametitle{Protocole DSA: La génération des clés}
Le signataire ALICE choisit:
\begin{enumerate}
\item un nombre premier $p$,  
\item $q$ un nombre premier diviseur de $p-1$,
\item $g$ appartenant à $\mathbb{Z} / p \mathbb{Z}$
\item une fonction de hachage $H$
\end{enumerate}
ALICE tire aléatoirement $x$ un élément de $\mathbb{Z} / q \mathbb{Z}$.\\
La clé publique = $(p,q,g,y)$\\
La clé secrète = $x$
\end{frame}
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\begin{frame}                           
\frametitle{Protocole DSA: La signature}
ALICE veut signer le message $M$.
\begin{enumerate}
\item choisir aléatoirement un élément $k$ de $\mathbb{Z} / q \mathbb{Z}$.
\item calculer $r=g^k \mod p \mod q$.
\item calculer $s=(H(M)+rx) \ast k^{-1} \mod q $
\end{enumerate}
$(r,s) = sign\{M\}$
\end{frame}
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\begin{frame}                           
\frametitle{Protocole DSA: La vérification}
\begin{enumerate}
\item $(r,s)$ appartient à $\mathbb{Z}q^2$
\item $r = g^{H(M)s^{-1}}y^{rs^{-1}} \mod p \mod q$
\end{enumerate}
\end{frame}
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\subsubsection{Signature et courbe elliptique}
\begin{frame}                           
\frametitle{Notion de courbe elliptique}
\includegraphics[scale=0.5]{./Images/courbeselliptiques.eps}\\
\begin{itemize}
\item Sur $\mathbb{Z} / p \mathbb{Z}$, $E = \lbrace(x,y) / y^2=x^3+Ax+B \rbrace $ avec $4 \ast A^3 + 27 \ast B^2 \neq 0 \mod p$.
\item $(E,+)$ est un groupe.
\item Problème du logarithme discret.
\end{itemize}
\end{frame}

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\begin{frame}                           
\frametitle{Comment signer avec une courbe elliptique-1}  
\begin{itemize}
\item Soit $M$ le message qu'Alice souhaite signer avec une courbe elliptique.
\item Soit $H$ une fonction de hachage cryptographique.
\item Considérons une courbe elliptique $E$ et G un point de cette courbe d'ordre $p$, où $p$ est un nombre premier. 
\end{itemize}
\end{frame}
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\begin{frame}                           
\frametitle{Comment signer avec une courbe elliptique-2}  
1) Génération des clés :
	\begin{itemize}
	\item Alice tire aléatoirement $s$ entre $1$ et $p-1$
	\item Alice calcule ensuite $Q=sG$
	\end{itemize}
$Q$ représente la clé publique et $s$ la clé secrète.
\end{frame}
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\begin{frame}
\frametitle{Comment signer avec une courbe elliptique-3}
2) Signature :
	\begin{itemize}
	\item Alice tire aléatoirement $k$ entre $1$ et $p-1$.
	\item Alice calcule ensuite les coordonnées $(i,j)$ du point $kG$.
	\item Il faut alors évaluer $x = i \mod p$.
	\item Alice calcule alors $y = k^{-1}(H(M) + sx)\mod p$
	\item Si $x$ ou $y$ prend une valeur nulle, alors il faut retirer un entier $k$ et refaire les calculs de $x$ et $y$.
	\end{itemize}
Le couple $(x,y)$ représente alors la signature du message.
\end{frame}
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\begin{frame}
\frametitle{Comment signer avec une courbe elliptique-4}
3)Vérification :\\
Lorsque Bob reçoit le message d'Alice, celui-ci doit alors:
	\begin{itemize}
	\item Vérifier que $(x,y)$ appartient à $\mathbb{Z}^2_p$
	\item Calculer $(H(M)y^{-1} \mod p)G + (xy^{-1}\mod p)Q$ pour retrouver $(i,j)$
	\item Vérifier que $x = i \mod p$
	\item Vérifier que Q est un bien un point de la courbe elliptique non nul et que $nQ=(0,0)$	
	\end{itemize}
\end{frame}
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\begin{frame}                           
\frametitle{Validité du schéma de signature}  

Démontrons que si $x=i \mod p$ alors on vérifie bien la signature.
\\
Pour cela calculons $(H(M)y^{-1} \mod p)G + (xy^{-1}\mod p)Q$.

$$= (H(M)y^{-1} \mod p)G + (xy^{-1}\mod p)sG$$

$$= ((H(M)y^{-1}+sxy^{-1})\mod p)G$$

$$= ((H(M)+sx)y^{-1}\mod p)G$$ 

Or $y = k^{-1}(H(M) + sx)\mod p$\\

$$(H(M)y^{-1} \mod p)G + (xy^{-1}\mod p)Q = ((H(M)+sx)k(H(M)+sx)^{-1})\mod p)G$$
$$= kG$$
Ainsi:
$$(H(M)y^{-1} \mod p)G + (xy^{-1}\mod p)Q = kG = (i,j)$$

\end{frame}

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\begin{frame}                           
\frametitle{Sécurité du schéma de signature}  
\begin{itemize}
\item Logarithme discret dans un groupe additif.
\item "$\rho$-Pollard" ou "pas de géant, pas de bébé" en $\mathcal{O}(\sqrt{p})$.
\item DSA classique :  nombre premier de l'ordre de $2^{1000}$ contre $2^{200}$ pour ECDSA. 
\end{itemize}
\end{frame}

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\subsubsection{Notion de certificat}
\begin{frame}                           
\frametitle{Utilité des certificats}  
Signature $ \Longrightarrow$ confiance ?\\
Non si aucune garantie que celui qui signe est bien ALICE.\\
Comment s'assurer que la clé publique reçue est bien celle d'ALICE?\\
\textbf{Solution : la certification}

\end{frame}
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\begin{frame}                           
\frametitle{Utilisation de certificats}  
\includegraphics[scale=0.5]{./Images/certification.eps}
\begin{itemize}
\item en signature ou authentification :\\
Alice veut être authentifiée par Bob ou lui envoyer un message signé.
\item en chiffrement :\\ 
Bob utilise un algorithme de chiffrement utilisant la clé publique de Alice pour chiffrer un message. 
\end{itemize}

\end{frame}


\section{ISO 7816/ protocole EMV}
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%				Slide EMV
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\begin{frame}                           
\frametitle{ISO 7816/ protocole EMV}

\begin{center}
\begin{itemize}
\item{généralités}
\item{l'avant EMV}
\end{itemize} 
\end{center}
 
\end{frame}

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%				Slide EMV
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\begin{frame} 
Déroulement de la transaction\\
\begin{itemize}

\item agents impliqués:\\
$T$ le terminal, $C$ la carte, $A$ comme Alice le détenteur de la carte, $PK$ la clé publique chiffrement et $SK$ la clé privé banque.\\
\item Les données manipulées:\\
le nom, le prénom, le numéro de carte, la date de validité, la signature de ces données avec la clé privée que nous noterons $\{data\}_{SK}$. Ces données ainsi que la signature sont enregistrées sur la carte au moment de son initialisation.\\
\item La transaction se déroule en hors-ligne, c'est à dire que tout se passe entre la carte et le terminal; seule la puce est utilisée pour faire les vérifications.
\end{itemize}

\end{frame}
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%				Slide EMV
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\begin{frame}
Déroulement de la transaction - première faille\\                       
\begin{enumerate}
\item $T \longrightarrow A $ : "authentification".
\item $C \longrightarrow T $ : $data$, $\{data \}_{SK}$. (le terminal utilise la clé publique $PK$ pour extraire "$data$" de la signature et les comparer aux "$data$" envoyées par la carte)
\item $T \longrightarrow A $ : code? 
\item $A \longrightarrow T $ : 1234 (par exemple)
\item $T \longrightarrow C $ : 1234
\item $C \longrightarrow T $ : ok.
\end{enumerate}

\end{frame}


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%				Slide EMV
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\begin{frame} 
Déroulement de la transaction - deuxième faille 

\begin{enumerate}
\item $T \longrightarrow A$ : "authentification"
\item $C \longrightarrow T$ : $data$ fabriquées, $\{data\}_{\{SK\}}$    le terminal valide cette étape
\item $T \longrightarrow A$ : code ?
\item $A \longrightarrow T$ : 7777 (par exemple)
\item $T \longrightarrow C$ : 7777
\item $C \longrightarrow T$ : ok.
\end{enumerate}

\end{frame}

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%				Slide EMV
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\begin{frame} 
La norme ISO 7816:protocole de transmission entre la puce de la carte et le lecteur. Ce protocole comporte 12 étapes.

\begin{itemize}
\item étape 1: sélection de l'application
\item étape 2: initier le traitement de l'application
\item étape 3:lecture des données de l'application
\item étape 4:traitement des restrictions
\item étape 5:authentification des données hors-ligne
\end{itemize}

\end{frame}


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%				Slide EMV
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\begin{frame} 


\begin{itemize}
\item étape 6: vérification de l'identité du détenteur de la carte
\item étape 7: gestion du risque par le terminal
\item étape 8: analyse de l'action par le terminal
\item étape 9: première analyse d'action par la carte
\item étape 10:autorisation de transaction en ligne
\item étape 11:seconde analyse de la carte
\item étape 12:traitement du script de l'émetteur
\end{itemize}

\end{frame}

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%				Slide EMV
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\begin{frame} 
Authentification avec mode SDA
\includegraphics[scale=0.4]{./images/SDA.eps}

\end{frame}

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%				Slide EMV
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\begin{frame} 
Authentification avec mode DDA
\includegraphics[scale=0.4]{./images/DDA.eps}

\end{frame}


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%				Slide 6
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\section{Modélisation}
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%				Slide 7
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\begin{frame}                           
\frametitle{Modélisation}  
\begin{itemize}
\item Phase d'apprentissage
\item Phase de test
\end{itemize}

\begin{center}
%\includegraphics[scale=0.3]{./Images/generation.eps}
\end{center}
\end{frame}

\section{BasicCard}
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%				Slide 8
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\begin{frame}                           
\frametitle{BasicCard}

\begin{itemize}
\item
\end{itemize}
\end{frame}



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%				Conclusion
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\section{Conclusion}

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%				Slide 9
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\begin{frame} 
                
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\end{document}
